Finite Group Databasebeta

← C2×#22586884 #31964708

Subgroups — C2×#22586884 #31964708

Order 18176.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2C2×#17985859C211yesmaximal, characteristic, complemented
3#22586884C218yesmaximal, direct product factor
4C2×#17985737C212yesmaximal, complemented
5C2×#17985734C212yesmaximal, complemented
6C2×#407×C71C211yesmaximal, characteristic, complemented
7C22×#11765865C411yescharacteristic, complemented
8#17986510—22no—
9#17986510—22no—
10#17986510C414yescomplemented
11#17986510C2214yes—
12C2×#14556111—21no—
13C2×#14556107C411yescharacteristic, complemented
14C2×#14556107C2211yescharacteristic
15C2×#14556102C411yescharacteristic, complemented
16C2×#14556102C2211yescharacteristic
17C2×#14556097—21no—
18C2×C4×C22⋊D71—21no—
19C2×C4×C22⋊D71—21no—
20C2×#14556087C411yescharacteristic, complemented
21C22×C4⋊Dic71—21no—
22C22×C22⋊Dic71—21no—
23#17985859—22no—
24#17985859C2214yescomplemented
25C2×C71⋊#95C2212yes—
26#17985737C2218yescomplemented
27#17985734C2218yescomplemented
28C24×C4×C71C2211yescharacteristic
29C22×M16×C71—21no—
30#407×C71—22no—
31#407×C71C2214yescomplemented
32C2×C22⋊C8×C71—21no—
33C2×C22⋊C8×C71C2212yes—
34C2×#11765865—22no—
35C2×#11765865—44no—
36C2×#11765865C2×C414yescomplemented
37C22×Dic142C2×C411yescharacteristic
38C22×D284C2×C411yescharacteristic
39C22×C22⋊D71—21no—
40C22×C22⋊D71—21no—
41C22×C22⋊D71—21no—
42C23×Dic71—22no—
43C22×C4×D71—21no—
44#14556111—24no—
45#14556111—24no—
46#14556107—22no—
47#14556107—21no—
48#14556107—21no—
49#14556107C2×C412yes—
50#14556107C2×C414yescomplemented
51#14556107C2312yes—
52#14556102—22no—
53#14556102—21no—
54#14556102—21no—
55#14556102C2×C412yes—
56#14556102C2×C414yescomplemented
57#14556102C2312yes—
58#14556097—24no—
59#14556097—24no—
60C4×C22⋊D71—24no—
61C4×C22⋊D71—24no—
62C4×C22⋊D71—24no—
63C4×C22⋊D71—24no—
64#14556087—22no—
65#14556087C2×C414yescomplemented
66C2×C4⋊Dic71—24no—
67C2×C4⋊Dic71—44no—
68C2×C4⋊Dic71C2×C411yescharacteristic
69C2×C4⋊Dic71C2×C411yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—21no—
72C2×C22⋊Dic71—21no—
73C2×C22⋊Dic71—21no—
74C2×C22⋊Dic71—21no—
75C2×C22⋊Dic71—24no—
76C2×C22⋊Dic71—44no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C71⋊#95—22no—
81C71⋊#95C2314yes—
82C25×C71D411yescharacteristic
83C23×C4×C71—44no—
84C23×C4×C71C2×C411yescharacteristic
85C23×C4×C71C2×C411yescharacteristic
86C23×C4×C71C2×C412yes—
87C23×C4×C71D411yescharacteristic
88C23×C4×C71D411yescharacteristic
89C23×C4×C71D412yes—
90C23×C4×C71D411yescharacteristic
91C23×C4×C71D412yes—
92C23×C4×C71C2311yescharacteristic
93C23×C4×C71C2312yes—
94C2×M16×C71—24no—
95C2×M16×C71—44no—
96C22×C8×C71—22no—
97C22⋊C8×C71—24no—
98C22⋊C8×C71—22no—
99C22⋊C8×C71C2314yes—
100C23×D71—21no—
101#11765865—48no—
102#11765865—48no—
103C2×Dic142—21no—
104C2×Dic142—42no—
105C2×Dic142C22×C412yes—
106C2×D284—21no—
107C2×D284—42no—
108C2×D284C22×C412yes—
109C2×C22⋊D71—24no—
110C2×C22⋊D71—22no—
111C2×C22⋊D71—24no—
112C2×C22⋊D71—22no—
113C2×C22⋊D71—44no—
114C2×C22⋊D71—44no—
115C2×C22⋊D71—42no—
116C2×C22⋊D71—42no—
117C22×Dic71—21no—
118C22×Dic71—21no—
119C22×Dic71—21no—
120C22×Dic71—24no—
121C22×Dic71—44no—
122C22×Dic71—44no—
123C22×Dic71—44no—
124C2×C4×D71—24no—
125C2×C4×D71—44no—
126C4⋊Dic71—21no—
127C4⋊Dic71—21no—
128C4⋊Dic71—48no—
129C4⋊Dic71C22×C412yes—
130C4⋊Dic71C22×C412yes—
131Dic71⋊C4—22no—
132Dic71⋊C4—24no—
133Dic71⋊C4—22no—
134C22⋊Dic71—24no—
135C22⋊Dic71—22no—
136C22⋊Dic71—24no—
137C22⋊Dic71—22no—
138C22⋊Dic71—48no—
139(C2×D71)⋊C4—22no—
140(C2×D71)⋊C4—24no—
141(C2×D71)⋊C4—22no—
142C4×Dic71—24no—
143C24×C71—22no—
144C24×C71—22no—
145C24×C71—42no—
146C24×C71—42no—
147C24×C71C22⋊C411yescharacteristic
148C24×C71C22⋊C411yescharacteristic
149C24×C71C22⋊C412yes—
150C24×C71C2×D411yescharacteristic
151C24×C71C2×D412yes—
152C22×C4×C71—24no—
153C22×C4×C71—24no—
154C22×C4×C71—24no—
155C22×C4×C71—44no—
156C22×C4×C71—44no—
157C22×C4×C71—44no—
158C22×C4×C71—44no—
159C22×C4×C71—42no—
160C22×C4×C71—42no—
161C22×C4×C71—42no—
162C22×C4×C71C22⋊C412yes—
163C22×C4×C71C22⋊C411yescharacteristic
164C22×C4×C71C22⋊C412yes—
165C22×C4×C71C22⋊C412yes—
166C22×C4×C71C22⋊C412yes—
167C22×C4×C71C22⋊C412yes—
168C22×C4×C71C22⋊C411yescharacteristic
169C22×C4×C71C22×C411yescharacteristic
170C22×C4×C71C22×C412yes—
171C22×C4×C71C22×C411yescharacteristic
172C22×C4×C71C22×C412yes—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D412yes—
177C22×C4×C71C2×D412yes—
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2411yescharacteristic
180M16×C71—48no—
181C2×C8×C71—24no—
182C2×C8×C71—44no—
183C23×C71—22no—
184C23×C71—22no—
185C23×C71—22no—
186C23×C71—22no—
187C23×C71—22no—
188C23×C71—22no—
189C23×C71—24no—
190C23×C71—22no—
191C23×C71—22no—
192C23×C71—22no—
193C23×C71—22no—
194C23×C71—42no—
195C23×C71—42no—
196C23×C71—42no—
197C23×C71—42no—
198C23×C71—48no—
199C23×C71—48no—
200C23×C71C2×C22⋊C412yes—
201C23×C71C2×C22⋊C412yes—
202C23×C71C2×C22⋊C411yescharacteristic
203C23×C71C2×C22⋊C411yescharacteristic
204C23×C71AGL(1,ℤ/8)12yes—
205C23×C71#13712yes—
206C23×C71C22×D411yescharacteristic
207C22×D71—22no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—41no—
223C2×C4×C71—42no—
224C2×C4×C71—41no—
225C2×C4×C71—42no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71#9914yes—
229C2×C4×C71C2×C22⋊C412yes—
230C2×C4×C71C2×C22⋊C411yescharacteristic
231C2×C4×C71C2×C22⋊C411yescharacteristic
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C23×C411yescharacteristic
234C2×C4×C71C22×D411yescharacteristic
235C22⋊D71—48no—
236C22⋊D71—48no—
237C22⋊D71—42no—
238C22⋊D71—42no—
239C22⋊D71—44no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—44no—
247C2×Dic71—44no—
248C2×Dic71—44no—
249C2×Dic71—88no—
250D284—42no—
251D284—42no—
252C4×D71—48no—
253Dic142—42no—
254Dic142—42no—
255C8×C71—44no—
256C22×C71—21no—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—24no—
261C22×C71—24no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—22no—
268C22×C71—44no—
269C22×C71—44no—
270C22×C71—48no—
271C22×C71—44no—
272C22×C71—44no—
273C22×C71C2×#9912yes—
274C22×C71#41712yes—
275C22×C71#41712yes—
276C22×C71#42112yes—
277C22×C71C22×C22⋊C411yescharacteristic
278C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C71C2×#13711yescharacteristic
280C2×D71—42no—
281C2×D71—42no—
282C2×D71—42no—
283C2×D71—84no—
284C4×C71—21no—
285C4×C71—21no—
286C4×C71—21no—
287C4×C71—21no—
288C4×C71—22no—
289C4×C71C2×#9914yes—
290Dic71—42no—
291Dic71—42no—
292Dic71—42no—
293Dic71—84no—
294C2×#1513—711nomaximal, Sylow
295C2×C71—21no—
296C2×C71—21no—
297C2×C71—22no—
298C2×C71—42no—
299C2×C71—42no—
300C2×C71#151314yescomplemented
301C2×C71C22×#9911yescharacteristic
302C2×C71C2×#41711yescharacteristic
303C2×C71C2×#42111yescharacteristic
304D71—82no—
305C2×#525—711no—
306C2×#407D7111yescharacteristic, complemented
307C2×#380—711no—
308#1513—718no—
309C2×#328—712no—
310C2×#327—712no—
311C71C2×#151311yescharacteristic, complemented, Sylow
312C22×#41—711no—
313C24×C4C2×D7111yescharacteristic
314C22×M16—21no—
315#525—714no—
316#525—714no—
317#525—1422no—
318#525—1422no—
319C2×#123—1421no—
320C2×C22⋊Q8—711no—
321C2×C22⋊Q8—711no—
322C2×#121—711no—
323C2×#121—711no—
324C2×#120—1421no—
325C2×C4×D4—1421no—
326C2×C4×D4—1421no—
327C2×#117—711no—
328C22×C4⋊C4—1421no—
329C22×C22⋊C4—1421no—
330#407—22no—
331#407C2×D7114yescomplemented
332C2×C22⋊C8—21no—
333C2×C22⋊C8C2×D7112yes—
334#380—714no—
335#380—1422no—
336C2×#95—712no—
337#328—718no—
338#327—718no—
339C25D28411yescharacteristic
340C2×#41—2844no—
341C2×#41—714no—
342C2×#41—1422no—
343C22×Q8—711no—
344C22×D4—711no—
345C22×D4—1421no—
346C22×D4—1421no—
347C22×D4—1421no—
348C23×C4—44no—
349C23×C4—1422no—
350C23×C4—1421no—
351C23×C4C4×D7111yescharacteristic
352C23×C4C4×D7111yescharacteristic
353C23×C4C4×D7112yes—
354C23×C4D28411yescharacteristic
355C23×C4D28412yes—
356C23×C4C22⋊D7111yescharacteristic
357C23×C4C22⋊D7111yescharacteristic
358C23×C4C22⋊D7112yes—
359C23×C4C22×D7111yescharacteristic
360C23×C4C22×D7112yes—
361C2×M16—24no—
362C2×M16—44no—
363C22×C8—22no—
364#123—1424no—
365#123—1424no—
366C22⋊Q8—712no—
367C22⋊Q8—714no—
368C22⋊Q8—712no—
369C22⋊Q8—1422no—
370C22⋊Q8—1421no—
371C22⋊Q8—1421no—
372#121—712no—
373#121—714no—
374#121—712no—
375#121—1422no—
376#121—1421no—
377#121—1421no—
378#120—1424no—
379#120—1424no—
380C4×D4—1424no—
381C4×D4—1424no—
382C4×D4—1424no—
383C4×D4—1424no—
384#117—714no—
385#117—1422no—
386C2×C4⋊C4—2844no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C22⋊C4—2844no—
393C2×C22⋊C4—1424no—
394C2×C22⋊C4—1421no—
395C2×C22⋊C4—1421no—
396C2×C22⋊C4—1421no—
397C2×C22⋊C4—1421no—
398C2×C22⋊C4—1421no—
399C2×C42—1421no—
400C22⋊C8—24no—
401C22⋊C8—22no—
402C22⋊C8C22×D7114yes—
403#95—714no—
404#95—1422no—
405C24—22no—
406C24—22no—
407C24—42no—
408C24—42no—
409C24—1421no—
410C24(C2×D71)⋊C411yescharacteristic
411C24(C2×D71)⋊C411yescharacteristic
412C24(C2×D71)⋊C412yes—
413C24C2×D28411yescharacteristic
414C24C2×D28412yes—
415#41—2848no—
416#41—2848no—
417C2×Q8—2842no—
418C2×Q8—712no—
419C2×Q8—1421no—
420C2×D4—2842no—
421C2×D4—2844no—
422C2×D4—2844no—
423C2×D4—2842no—
424C2×D4—2842no—
425C2×D4—712no—
426C2×D4—1422no—
427C2×D4—1424no—
428C2×D4—1422no—
429C2×D4—1424no—
430C2×D4—1421no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—2844no—
442C22×C4—2844no—
443C22×C4—2844no—
444C22×C4—2844no—
445C22×C4—1424no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4—1424no—
450C22×C4(C2×D71)⋊C412yes—
451C22×C4(C2×D71)⋊C412yes—
452C22×C4(C2×D71)⋊C412yes—
453C22×C4(C2×D71)⋊C411yescharacteristic
454C22×C4(C2×D71)⋊C412yes—
455C22×C4(C2×D71)⋊C412yes—
456C22×C4(C2×D71)⋊C411yescharacteristic
457C22×C4C2×C4×D7112yes—
458C22×C4C2×C4×D7111yescharacteristic
459C22×C4C2×C4×D7112yes—
460C22×C4C2×C4×D7111yescharacteristic
461C22×C4C2×C22⋊D7112yes—
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7112yes—
465C22×C4C2×D28412yes—
466C22×C4C2×D28411yescharacteristic
467C22×C4C23×D7111yescharacteristic
468M16—48no—
469C2×C8—24no—
470C2×C8—44no—
471C4⋊C4—2848no—
472C4⋊C4—712no—
473C4⋊C4—712no—
474C4⋊C4—1422no—
475C4⋊C4—1422no—
476C4⋊C4—1424no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C22⋊C4—2848no—
480C22⋊C4—1422no—
481C22⋊C4—1424no—
482C22⋊C4—1424no—
483C22⋊C4—1422no—
484C22⋊C4—1422no—
485C22⋊C4—1422no—
486C22⋊C4—1424no—
487C42—1424no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—48no—
505C23—2842no—
506C23—2842no—
507C23—2842no—
508C23—1422no—
509C23C2×(C2×D71)⋊C412yes—
510C23C2×(C2×D71)⋊C412yes—
511C23C2×(C2×D71)⋊C411yescharacteristic
512C23C2×(C2×D71)⋊C411yescharacteristic
513C23#1455615612yes—
514C23#1455616112yes—
515C23C22×D28411yescharacteristic
516Q8—2842no—
517Q8—2842no—
518D4—2842no—
519D4—2842no—
520D4—2848no—
521D4—2848no—
522D4—2844no—
523D4—2842no—
524D4—2842no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—2842no—
543C2×C4—2842no—
544C2×C4—2842no—
545C2×C4—2848no—
546C2×C4—2844no—
547C2×C4—2844no—
548C2×C4—2844no—
549C2×C4—5688no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4#1455604214yes—
554C2×C4C2×(C2×D71)⋊C411yescharacteristic
555C2×C4C2×(C2×D71)⋊C412yes—
556C2×C4C2×(C2×D71)⋊C411yescharacteristic
557C2×C4C2×(C2×D71)⋊C412yes—
558C2×C4C22×C4×D7111yescharacteristic
559C2×C4C22×C22⋊D7111yescharacteristic
560C8—44no—
561C22—24no—
562C22—24no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—2842no—
579C22—2842no—
580C22—2842no—
581C22—5684no—
582C22C2×#1455604212yes—
583C22#1798600112yes—
584C22#1798600112yes—
585C22#1798601512yes—
586C22C22×(C2×D71)⋊C411yescharacteristic
587C22C2×#1455615611yescharacteristic
588C22C2×#1455616111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—2842no—
595C4—2842no—
596C4—2842no—
597C4—5684no—
598C4C2×#1455604214yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—42no—
603C2—42no—
604C2—5682no—
605C2#2258688414yesdirect product factor
606C2C22×#1455604211yescharacteristic
607C2C2×#1798600111yescharacteristic
608C2C2×#1798601511yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#407×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC2×#407D71
hypocenterC71C2×#1513
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#17985859C2semidirect
3. #22586884C2direct (N × H)
4. C2×#17985737C2semidirect
5. C2×#17985734C2semidirect
6. C2×#407×C71C2semidirect
7. C22×#11765865C4semidirect
10. #17986510C4semidirect
13. C2×#14556107C4semidirect
15. C2×#14556102C4semidirect
20. C2×#14556087C4semidirect
24. #17985859C22semidirect
26. #17985737C22semidirect
27. #17985734C22semidirect
31. #407×C71C22semidirect
36. C2×#11765865C2×C4semidirect
50. #14556107C2×C4semidirect
56. #14556102C2×C4semidirect
65. #14556087C2×C4semidirect
300. C2×C71#1513semidirect
306. C2×#407D71semidirect
311. C71C2×#1513semidirect
331. #407C2×D71semidirect
605. C2#22586884direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179865102
C2×#179858592
#225868842
C2×#179857372
C2×#179857342
C2×#407×C712
C2×#151371